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Units and Measurements question

2019 · 10 Apr · Shift 2 · Q64
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Units and Measurements question

2019 · 10 Apr · Shift 2 · Q64

JEE MainPhysicsUnits and MeasurementsMCQ+4 / −1
In the formula X = 5YZ2 , X and Z have dimensions of capacitance and magnetic field, respectively. What are the dimensions of Y in SI units?
  1. A
    [M–3L–2T8A4]
  2. B
    [M–2L–2T6A3]
  3. C
    [M–1L–2T4A2]
  4. D
    [M–2L0 T–4A–2]
View written solutionFree

Correct answer: A

  1. Given relation

    X=5YZ2X = 5YZ^2X=5YZ2

    The constant 555 is dimensionless, so:

    [X]=[Y][Z]2[X] = [Y][Z]^2[X]=[Y][Z]2

    Hence,

    [Y]=[X][Z]2[Y] = \frac{[X]}{[Z]^2}[Y]=[Z]2[X]​

  2. Dimensions of XXX (capacitance)

    Capacitance is

    C=QVC = \frac{Q}{V}C=VQ​

    Now,

    [Q]=[AT][Q] = [AT][Q]=[AT]

    and

    [V]=workcharge=[ML2T−2][AT]=[ML2T−3A−1][V] = \frac{\text{work}}{\text{charge}} = \frac{[ML^2T^{-2}]}{[AT]} = [ML^2T^{-3}A^{-1}][V]=chargework​=[AT][ML2T−2]​=[ML2T−3A−1]

    Therefore,

    [X]=[C]=[AT][ML2T−3A−1]=[M−1L−2T4A2][X] = [C] = \frac{[AT]}{[ML^2T^{-3}A^{-1}]} = [M^{-1}L^{-2}T^4A^2][X]=[C]=[ML2T−3A−1][AT]​=[M−1L−2T4A2]

  3. Dimensions of ZZZ (magnetic field)

    From Lorentz force,

    F=qvBF = qvBF=qvB

    so

    [B]=[F][q][v][B] = \frac{[F]}{[q][v]}[B]=[q][v][F]​

    [F]=[MLT−2],[q]=[AT],[v]=[LT−1][F] = [MLT^{-2}], \quad [q] = [AT], \quad [v] = [LT^{-1}][F]=[MLT−2],[q]=[AT],[v]=[LT−1]

    Thus,

    [Z]=[B]=[MLT−2][AT][LT−1]=[MT−2A−1][Z] = [B] = \frac{[MLT^{-2}]}{[AT][LT^{-1}]} = [MT^{-2}A^{-1}][Z]=[B]=[AT][LT−1][MLT−2]​=[MT−2A−1]

  4. Compute dimensions of YYY

    [Y]=[X][Z]2=[M−1L−2T4A2][MT−2A−1]2[Y] = \frac{[X]}{[Z]^2} = \frac{[M^{-1}L^{-2}T^4A^2]}{[MT^{-2}A^{-1}]^2}[Y]=[Z]2[X]​=[MT−2A−1]2[M−1L−2T4A2]​

    First,

    [Z]2=[M2T−4A−2][Z]^2 = [M^2T^{-4}A^{-2}][Z]2=[M2T−4A−2]

    Therefore,

    [Y]=[M−1L−2T4A2]⋅[M−2T4A2][Y] = [M^{-1}L^{-2}T^4A^2] \cdot [M^{-2}T^4A^2][Y]=[M−1L−2T4A2]⋅[M−2T4A2]

    [Y]=[M−3L−2T8A4][Y] = [M^{-3}L^{-2}T^8A^4][Y]=[M−3L−2T8A4]

  5. Match with options

    [Y]=[M−3L−2T8A4][Y] = [M^{-3}L^{-2}T^8A^4][Y]=[M−3L−2T8A4]

    This matches Option A.

Final Answer: A

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